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# From closed categories to closed multicategories

In document 2. Closed categories (Stránka 31-45)

In this section we prove our main result.

5.1. Theorem.The Cat-functor U :ClMulticatu →ClCat is a Cat-equivalence.

We have to prove that U is bijective on 1-morphisms and 2-morphisms, and that it is essentially surjective; the latter means that for each closed category V there is a closed multicategory with a unit object such that its underlying closed category is isomorphic (as a closed category) to V.

5.2. The surjectivity of U on 1-morphismsLetCand Dbe closed multicategories with unit objects. Denote their underlying closed categories by the same symbols. Let Φ = (φ,φ, φˆ 0) : C → D be a closed functor. We are going to define a multifunctor

F : C→ Dwhose underlying closed functor is Φ. Define F X =φX, for each X ∈ ObC. For each Y ∈ObC, the map F;Y :C(;Y)→D(;φY) is defined via the diagram

C(;Y) D(;φY)

C(1;Y) D(φ1;φY) D(1;φY)

F;Y

C(u;1) D(u;1)

φ D0;1)

Recall that for a morphism f : ()→Y we denote by f :1→Y a unique morphism such that u·f =f. Then the commutativity in the above diagram means that

F f =

()−→u 1 φ

−→0 φ1−−→φ(f) φY

, (5.1)

for each f : ()→Y. For n≥1 and X1, . . . , Xn, Y ∈ObC, the map FX1,...,Xn;Y :C(X1, . . . , Xn;Y)→D(φX1, . . . , φXn;φY) is defined inductively by requesting the commutativity in the diagram

C(X2, . . . , Xn;C(X1;Y)) D(φX2, . . . , φXn;φC(X1;Y))

D(φX2, . . . , φXn;D(φX1;φY))

C(X1, . . . , Xn;Y) D(φX1, . . . , φXn;φY)

FX2,...,Xn;C(X1;Y)

ϕC

D(1; ˆφ)

ϕD

FX1,...,Xn;Y

(5.2)

5.3. Lemma.The following diagram commutes

C(;C(X;Y)) D(;φC(X;Y)) D(;D(φX;φY))

C(X;Y) D(φX;φY)

F;C(X;Y) D(; ˆφ)

ϕC ϕD

φ

In particular, FX;YX,Y :C(X;Y)→D(φX;φY).

Proof.Equivalently, the exterior of the diagram

C(;C(X;Y)) D(;φC(X;Y)) D(;D(φX;φY))

C(1;C(X;Y)) D(φ1;φC(X;Y)) D(1;φC(X;Y)) D(1;D(φX;φY))

C(X;Y) D(φX;φY)

F;C(X;Y) D(; ˆφ)

C(u;1)

D(u;1) D(u;1)

φ D0;1) D(1; ˆφ)

γ γ

φ

C)−1 D)−1

commutes. The upper pentagon is the definition of F;C(X;Y). The bottom hexagon com-mutes. Indeed, taking f ∈C(X;Y) and tracing it along the left-top path yields

φ0·φ(jX)·φC(1;f)·φˆ=φ0·φ(jX)·φˆ·D(1;φ(f)) (naturality of ˆφ)

=jφX ·D(1;φ(f)), (axiom CF1) which is precisely the image of f along the bottom-right path.

5.4. Lemma.For each f : () →Y and Z ∈ObC, the diagram φC(Y;Z) φC(;Z) φZ

D(φY;φZ) D(;φZ) φZ

φC(f;1)

φˆ

D(F f;1)

commutes.

Proof.By definition,

F f =

()−→u 1−→φ0 φ1−−→φ(f) φY . The diagram

φC(Y;Z) φC(1;Z) φC(;Z) φZ

D(φY;φZ) D(φ1;φZ) D(;φZ) φZ

φC(f;1) φC(u;1)

φˆ φˆ

D(φ(f);1) D(u·φ0;1) φC(f;1)

D(F f;1)

commutes. Indeed, the left square commutes by the naturality of ˆφ, while the commuta-tivity of the right square is a consequence of the axiom CF2, see (4.1).

With the notation of Lemma 3.14, we can rewrite the commutativity condition in diagram (5.2) as a recursive formula for the multigraph morphism F:

F f =ϕD(F((ϕC)−1(f))·φ) =ˆ ϕD(Fhfi ·φ),ˆ for each f :X1, . . . , Xn →Y with n≥1, or equivalently

hF fi=

φX2, . . . , φXn Fhfi

−−→φC(X1;Y)−→φˆ D(φX1;φY)

. (5.3)

5.5. Lemma.For each X, Y, Z ∈ObC, the diagram

φC(X;Y), φC(Y;Z) φC(X;Z)

D(φX;φY),D(φY;φZ) D(φX;φZ)

F µC

φ,ˆφˆ φˆ

µD

commutes.

Proof.It suffices to prove the equation

hF µC·φiˆ =h( ˆφ,φ)ˆ ·µDi.

By Lemma 3.14,(c), the left hand side is equal to

φC(Y;Z)−−−→hF µCi D(φC(X;Y);φC(X;Z))−−−→D(1; ˆφ) D(φC(X;Y);D(φX;φZ)), while the right hand side is equal to

φC(Y;Z)−→φˆ D(φY;φZ)−−→Di D(D(φX;φY);D(φY;φZ))−−−→D( ˆφ;1) D(φC(X;Y);D(φX;φZ)) by Lemma 3.14,(b). Note that hµDi= (ϕD)−1D) =LφX. Furthermore, by (5.3),

hF µCi=

φC(Y;Z)−−−→φhµCi φC(C(X;Y);C(X;Z))−→φˆ D(φC(X;Y);φC(X;Z))

=

φC(Y;Z)−−→φLX φC(C(X;Y);C(X;Z))−→φˆ D(φC(X;Y);φC(X;Z)) , therefore the equation in question is simply the axiom CF3.

5.6. Proposition. The multigraph morphism F : C → D is a multifunctor, and its underlying closed functor is Φ.

Proof.Trivially, F preserves identities since so does φ. Let us prove that F preserves composition. The proof is in three steps.

5.7. Lemma.F preserves composition of the form X1, . . . , Xk

f

→Y −→g Z.

Proof. The proof is by induction on k. There is nothing to prove in the case k = 1.

Suppose that k = 0 and we are given composable morphisms ()−→f X −→g Y.

Then since u·f g = f · g = (u ·f) ·g = u·(f ·g), it follows that f·g = f ·g. By formula (5.1),

F(f ·g) =u·φ0·φ(f·g) = u·φ0·φ(f ·g) =u·φ0·φ(f)·φ(g) =F f·F g.

Suppose that k >1. Then

hF(f·g)i=Fhf·gi ·φˆ (formula (5.3))

=F(hfi ·C(1;g))·φˆ (Lemma 3.14,(c))

=Fhfi ·φC(1;g)·φˆ (induction hypothesis)

=Fhfi ·φˆ·D(1;φ(g)) (naturality of ˆφ)

=hF fi ·D(1;F g) (formula (5.3))

=hF f·F gi, (Lemma 3.14,(c)) and induction goes through.

5.8. Lemma.F preserves composition of the form

X11, . . . , X1k1, X21, . . . , X2k2 −−−→f1,f2 Y1, Y2 −→g Z

.

Proof.The proof is by induction on k1. If k1 = 0, then by Lemma 3.14,(a), (f1, f2)·g =

X21, . . . , X2k2f2 Y2

−→hgi C(Y1;Z)−−−−→C(f1;1) C(;Z) =Z , therefore

F((f1, f2)·g) =F f2·φhgi ·φC(f1; 1) (Lemma 5.7)

=F f2·φhgi ·φˆ·D(φ(f1); 1) (Lemma 5.4)

=F f2· hF gi ·D(F f1; 1) (formula (5.3))

= (F f1, F f2)·F g. (Lemma 3.14,(a)) If k1 = 1, then by Lemma 3.14,(b),

h(f1, f2)·gi=

X21, . . . , X2k2f2 Y2

−→hgi C(Y1;Z)−−−−→C(f1;1) C(X11;Z) ,

therefore

hF((f1, f2)·g)i=Fh(f1, f2)·gi ·φˆ (formula (5.3))

=F f2 ·φhgi ·φC(f1; 1)·φˆ (Lemma 5.7)

=F f2 ·φhgi ·φˆ·D(φ(f1); 1) (naturality of ˆφ)

=F f2 · hF gi ·D(F f1; 1) (formula (5.3))

=h(F f1, F f2)·F gi, (Lemma 3.14,(b))

and henceF((f1, f2)·g) = (F f1, F f2)·F g. Suppose thatk1 >1. Then by Lemma 3.14,(c) h(f1, f2)·gi is equal to the composite

X12, . . . , X1k1, X21, . . . , X2k2 −−−−→hf1i,f2 C(X11;Y1), Y2 1,hgi

−−→C(X11;Y1),C(Y1;Z)−→µC C(X11;Z), therefore

hF((f1, f2)·g)i=Fh(f1, f2)·gi ·φˆ (formula (5.3))

= (Fhf1i, F f2)·F((1,hgi)µC)·φˆ (induction hypothesis)

= (Fhf1i, F f2)·(1, Fhgi)·F µC·φˆ (casek1 = 1)

= (Fhf1i, F f2)·(1, Fhgi)·( ˆφ,φ)ˆ ·µD (Lemma 5.5)

= (Fhf1i ·φ, F fˆ 2)·(1, Fhgi ·φ)ˆ ·µD

= (hF f1i, F f2)·(1,hF gi)·µD (formula (5.3))

=h(F f1, F f2)·F gi, (Lemma 3.14,(c)) hence F((f1, f2)·g) = (F f1, F f2)·F g, and the lemma is proven.

5.9. Lemma.F preserves composition of the form

X11, . . . , X1k1, . . . , Xn1, . . . , Xnkn −−−−→f1,...,fn Y1, . . . , Yn

g

→Z. (5.4)

Proof.The proof is by induction on n, and for a fixed n by induction on k1. We have worked out the cases n = 1 andn = 2 explicitly in Lemmas 5.7 and 5.8. Assume that F preserves an arbitrary composition of the form

U11, . . . , U1l1, . . . , Un−11 , . . . , Un−1ln−1 −−−−−−→p1,...,pn−1 V1, . . . , Vn−1

q

→W,

and suppose we are given composite (5.4). We do induction on k1. If k1 = 0, then by Lemma 3.14,(a) (f1, . . . , fn)·g is equal to the composite

X21, . . . , X2k2, . . . , Xn1, . . . , Xnkn −−−−→f2,...,fn Y2, . . . , Yn

−→hgi C(Y1;Z)−−−−→C(f1;1) C(;Z) =Z,

therefore

F((f1, . . . , fn)·g) = (F f2, . . . , F fn)·F(hgi ·C(f1; 1)) (induction hypothesis)

= (F f2, . . . , F fn)·(Fhgi ·φC(f1; 1)) (Lemma 5.7)

= (F f2, . . . , F fn)·(Fhgi ·φˆ·D(φ(f1); 1)) (Lemma 5.4)

= (F f2, . . . , F fn)·(hF gi ·D(F f1; 1)) (formula (5.3))

= (F f1, . . . , F fn)·F g. (Lemma 3.14,(a))

Suppose thatk1 = 1. Then by Lemma 3.14,(b) h(f1, . . . , fn)·gi is equal to the composite X21, . . . , X2k2, . . . , Xn1, . . . , Xnkn −−−−→f2,...,fn Y2, . . . , Yn

−→hgi C(Y1;Z)−−−−→C(f1;1) C(X11;Z), therefore

hF((f1, . . . , fn)·g)i=Fh(f1, . . . , fn)·gi ·φˆ (formula (5.3))

= (F f2, . . . , F fn)·F(hgi ·C(f1; 1))·φˆ (induction hypothesis)

= (F f2, . . . , F fn)·Fhgi ·φC(f1; 1)·φˆ (Lemma 5.7)

= (F f2, . . . , F fn)·Fhgi ·φˆ·D(φ(f1); 1) (naturality of ˆφ)

= (F f2, . . . , F fn)· hF gi ·D(F f1; 1) (formula (5.3))

=h(F f1, . . . , F fn)·F gi, (Lemma 3.14,(b))

and hence F((f1, . . . , fn) · g) = (F f1, . . . , F fn) · F g. Suppose that k1 > 1, then by Lemma 3.14,(c) h(f1, . . . , fn)·gi is equal to the composite

X12, . . . , X1k1, X21, . . . , X2k2, . . . , Xn1, . . . , Xnkn −−−−−−−→hf1i,f2,...,fn C(X11;Y1), Y2, . . . , Yn 1,hgi

−−−−−−−→C(X11;Y1),C(Y1;Z)

µC

−−−−−−−→C(X11;Z), therefore

hF((f1, . . . , fn)·g)i=Fh(f1, . . . , fn)·gi ·φˆ (formula (5.3))

= (Fhf1i, F f2, . . . , F fn)·F((1,hgi)µC)·φˆ (induction hypothesis)

= (Fhf1i, F f2, . . . , F fn)·(1, F[g])·F µC·φˆ (Lemma 5.8)

= (Fhf1i, F f2, . . . , F fn)·(1, Fhgi)·( ˆφ,φ)ˆ ·µD (Lemma 5.5)

= (Fhf1i ·φ, F fˆ 2, . . . , F fn)·(1, Fhgi ·φ)ˆ ·µD

= (hF f1i, F f2, . . . , F fn)·(1,hF gi)·µD (formula (5.3))

=h(F f1, . . . , F fn)·F gi, (Lemma 3.14,(c)) hence F((f1, . . . , fn)·g) = (F f1, . . . , F fn)·F g, and induction goes through.

Thus we have proven that F : C → D is a multifunctor. By construction, its un-derlying functor is φ. Furthermore, the closing transformation FX;Y coincides with φˆX,Y : φC(X;Y) → D(φX;φY). Indeed, we first observe that FX,Y = hF evCi, where closed functor is Φ. The proposition is proven.

5.10. The injectivity of U on 1-morphisms The following proposition shows that the Cat-functor U is injective on 1-morphisms.

5.11. Proposition. Let F, G : C → D be multifunctors between closed multicategories with unit objects. Suppose that F and G induce the same closed functor Φ = (φ,φ, φˆ 0) between the underlying closed categories. Then F =G.

Proof. By assumption, the underlying functors of the multifunctors F and G are the same and are equal to the functor φ. Let us prove that F f = Gf, for each f : X1, . . . , Xn → Y. The proof is by induction on n. There is nothing to prove if n = 1.

Suppose thatn = 0, i.e.,f is a morphism ()→Y. Then since F and Gare multifunctors, F f =F(u·f) =F u·F f , Gf =G(u·f) =Gu·Gf .

Since F and G coincide on morphisms with one source object, it follows that F f =Gf. Furthermore,

hence F f =Gf. The induction step follows from the commutative diagram C(X2, . . . , Xn;C(X1;Y)) D(φX2, . . . , φXn;φC(X1;Y))

and a similar diagram for G, which are particular cases of Proposition 3.20.

5.12. The bijectivity of U on 2-morphismsThe following proposition implies that U is bijective on 2-morphisms.

5.13. Proposition. Let F, G : C → D be multifunctors between closed multicategories with unit objects. Denote by Φ = (φ,φ, φˆ 0) and Ψ = (ψ,ψ, ψˆ 0) the corresponding closed functors. Let r: Φ→Ψbe a closed natural transformation. Thenr is also a multinatural transformation F →G:C→D.

Proof.We must prove that, for each f :X1, . . . , Xn→Y, the equation F f·rY = (rX1, . . . , rXn)·Gf

holds true. The proof is by induction onn. Suppose thatn= 0, and thatf is a morphism ()→Y. The axiom CN1

1 φ

−→0 F1−r1 G1

0 implies

()−→F u F1−r1 G1

=Gu.

It follows that

F f ·rY =F u·F f·rY =F u·r1·Gf =Gu·Gf =Gf,

where the second equality is due to the naturality of r. There is nothing to prove in the case n = 1. Suppose that n >1. It suffices to prove that

hF f·rYi=h(rX1, . . . , rXn)·Gfi:F X2, . . . , F Xn→D(F X1;GY).

By Lemma 3.14,(c), the left hand side expands out as hF fi · D(1;rY), which by for-mula (5.3) is equal to Fhfi ·φˆ·D(1;rY). By Lemma 3.14,(b), the right hand side of the equation in question is equal to (rX2, . . . , rXn)· hGfi ·D(rX1; 1), which by formula (5.3) is equal to (rX2, . . . , rXn)·Ghfi ·ψˆ·D(rX1; 1). By the induction hypothesis, the latter is equal toFhfi ·rC(X1;Y)·ψˆ·D(rX1; 1). The required equation follows then from the axiom CN2.

5.14. The essential surjectivity of U Let us prove that for each closed categoryV there is a closed multicategory V with a unit object whose underlying closed category is isomorphic to V. First of all, notice that by Theorem 2.19 we may (and we shall) assume in what follows that V is a closed category in the sense of Eilenberg and Kelly; i.e., that V is equipped with a functor V : V → S such that VV(−,−) = V(−,−) : Vop ×V → S and the axiom CC5’ is satisfied. In particular, we can use the whole theory of closed categories developed in  without any modifications. We are now going to construct a closed multicategory Vwith a unit object whose underlying closed category is isomorphic toV. The construction is based on ideas of Laplaza’s paper .

We begin by recalling that for each object X of the category V one can assign a V-functorLX :V→V, and for eachf ∈VV(X, Y) =V(X, Y) there is a uniqueV-natural

transformation Lf : LY → LX : V → V such that (V(Lf)Y)1Y = f, see Examples 2.13, 2.15, 2.21, or [2, Section 9]. Moreover, by [2, Proposition 9.2] the assignments X 7→LX and f 7→ Lf determine a fully faithful functor from the category Vop to the category V-Cat(V,V) of V-functors V → V and their V-natural transformations. For us it is more convenient to write it as functor from V to V-Cat(V,V)op. Note that the latter category is strict monoidal with the tensor product given by composition of V-functors.

More precisely, the tensor product ofF and Gin the given order isF G=F ·G=G◦F. Consider the multicategory associated withV-Cat(V,V)op (see Example 3.3) and consider its full submulticategory whose objects areV-functorsLX,X ∈ObV. That is, in essence, our V. More precisely, ObV= ObVand

V(X1, . . . , Xn;Y) =V-Cat(V,V)op(LX1 ·. . .·LXn, LY)

=V-Cat(V,V)(LY, LXn◦ · · · ◦LX1).

Identities and composition coincide with those of the multicategory associated with the strict monoidal categoryV-Cat(V,V)op. Note that by Proposition 2.20 there is a bijection

Γ :V(X1, . . . , Xn;Y)→(V ◦LXn◦ · · · ◦LX1)Y, f 7→(V fY)1Y.

5.15. Theorem. The multicategory V is closed and has a unit object. The underlying closed category of V is isomorphic to V.

Proof. First, let us check that the multicategory V is closed. By Proposition 3.9, it suffices to prove that for each pair of objects X and Z there exist an internal Hom-object V(X;Z) and an evaluation morphism evVX;Z :X,V(X;Z)→Z such that the map

ϕ :V(Y1, . . . , Yn;V(X;Z))→V(X, Y1, . . . , Yn;Z), f 7→(1X, f)·evVX;Z,

is bijective, for each sequence of objects Y1, . . . , Yn. We set V(X;Z) = V(X, Z). The evaluation map evVX;Z : X,V(X;Z) → Z is by definition a V-natural transformation LZ →LV(X,Z)◦LX. We define it by requesting (V(evVX;Z)Z)1Z = 1V(X,Z) (we extensively use the representation theorem for V-functors in the form of Proposition 2.20). Let us check that the map ϕ is bijective. Note that the codomain of ϕ identifies via the map Γ with the set (V ◦LYn ◦ · · · ◦LY1 ◦LX)Z, and that the domain of ϕ identifies via Γ with the set

(V ◦LYn◦ · · · ◦LY1)V(X, Z) = (V ◦LYn◦ · · · ◦LY1 ◦LX)Z.

The bijectivity of ϕ follows readily from the diagram

V(Y1, . . . , Yn;V(X;Z)) V(X, Y1, . . . , Yn;Z)

(V ◦LYn◦ · · · ◦LY1 ◦LX)Z

ϕ

Γ Γ

whose commutativity we are going to establish. Take an f ∈V(Y1, . . . , Yn;V(X;Z)), i.e.,

Thus we conclude that Vis a closed multicategory.

Let us check that1∈ObVis a unit object ofV. By definition, a morphismu: ()→1 is a V-natural transformation L1 → Id. We let it be equal to i−1, which is a V-natural transformation by [2, Proposition 8.5]. Then for each object X of V holds

V(u; 1) = (u,1)·evV1;X :V(1;X)→X, i.e., V(u; 1) is the V-natural transformation

LX ev

V

−−−→1;X LV(1;X)◦L1 L

V(1,X)u

−−−−−→LV(1,X).

We claim that it coincides with Li−1X and hence is invertible. Indeed, applying Γ to the above composite we obtain

Let us now describe the underlying closed category of the closed multicategory V. Its objects are those of V, and for each pair of objects X and Y the set of morphisms from X to Y is V(X;Y) = V-Cat(V,V)(LY, LX). The unit object is 1 and the internal Hom-object V(X;Y) coincides with V(X, Y). For each object X, the identity morphism 1VX : () → V(X;X), i.e., a V-natural transformation LV(X,X) → Id, is found from the Applying Γ to both sides we find that

V((1VXLX)◦evVX;X)X

HereV(1VX)V(X,X) :V(V(X, X),V(X, X))→V(X, X). The morphismjX of the underlying closed category of V is a V-natural transformation LV(X,X) → L1; it is found from the equation

LV(X,X) −→jX L1−→u Id

= 1VX. Applying Γ to both sides we obtain

V(u◦jX)V(X,X)

1V(X,X) =V(1VX)V(X,X)1V(X,X), i.e.,

V i−1V(X,X)

V(jX)V(X,X)1V(X,X)

= 1X, or equivalently

(V(jX)V(X,X))1V(X,X) = (V iV(X,X))1X =jX,

where the last equality is the axiom CC5’. Therefore, jX = LjX : LV(X,X) → L1. It also follows by construction that iX for the underlying closed category of the closed multicategory V is (V(u; 1))−1 = (Li−1X )−1 =LiX.

Let us compute the morphism LXY Z : V(Y;Z) → V(V(X;Y);V(X;Z)). First note that evVX;Y : X,V(X;Y) → Y is the V-natural transformation LY → LV(X,Y)◦LX with components

(evVX;Y)Z =LXY Z :V(Y, Z)→V(V(X, Y),V(X, Z)).

In other words, evVX;Y =LXY,−. Indeed, applying Γ to both side of the equation in question we obtain an equivalent equation

(V(evVX;Y)Y)1Y = (V LXY Y)1Y.

Since V LX =V(X,−), it follows that (V LXY Y)1Y = 1V(X,Y), so that the above equation is just the definition of evVX;Y.

The morphism LXY Z : V(Y;Z) → V(V(X;Y);V(X;Z)) is uniquely determined by requesting that the diagram

X,V(X;Y),V(Y;Z) X,V(X;Y),V(V(X;Y);V(X;Z))

X,V(X;Z)

Y,V(Y;Z) Z

1,1,LXY Z

1,evVV(X;Y);V(X;Z)

evVX;Z evVX;Y,1

evVY;Z

in the multicategory V, or equivalently the diagram

LZ LV(X,Z)◦LX

LV(V(X,Y),V(X,Z))◦LV(X,Y)◦LX

LV(Y,Z)◦LY LV(Y,Z)◦LV(X,Y)◦LX

evVX;Z

evVV(X,Y);V(X,Z)LX

LXY ZLV(X,Y)LX evVY;Z

LV(Y,Z)evVX;Y

in the category V-Cat(V,V) commute. Applying Γ to both paths in the latter diagram we obtain

V(LXY Z)V(V(X,Y),V(X,Z))

V(evVV(X,Y);V(X,Z))V(X,Z)

V(evVX;Z)Z

1Z

=V(V(Y, Z),(evVX;Y)Z)(V(evVY;Z)Z)1Z, or equivalently

V(LXY Z)V(V(X,Y),V(X,Z))

1V(V(X,Y),V(X,Z)) = (evVX;Y)Z =LXY Z. In other words, LXY Z for the underlying closed category of V isLLXY Z.

Let us denote the underlying closed category of the multicategory V by the same symbol. There is a closed functor (L,1,1) : V → V, where L : V → V is given by X 7→X, f 7→Lf, and the morphisms V(X, Y)→V(X;Y) and 1→1 are the identities.

The axioms CF1–CF3 follow readily from the above description of the closed category V. Clearly, the functor Lis an isomorphism. The theorem is proven.

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In document 2. Closed categories (Stránka 31-45)

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